We show that if $p=O(1/n)$, then the Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$ with high probability admits a canonical labeling computable in time $O(n\log n)$. Combined with the previous results on the canonization of random graphs, this implies that $G(n,p)$ with high probability admits a polynomial-time canonical labeling whatever the edge probability function $p$. Our algorithm combines the standard color refinement routine with simple post-processing based on the classical linear-time tree canonization. Noteworthy, our analysis of how well color refinement performs in this setting allows us to complete the description of the automorphism group of the 2-core of $G(n,p)$.

翻译：我们证明，当 $p=O(1/n)$ 时，Erdős-Rényi 随机图 $G(n,p)$ 以高概率存在一个可在 $O(n\log n)$ 时间内计算的标准标记。结合先前关于随机图标准化的研究结果，这表明无论边概率函数 $p$ 如何，$G(n,p)$ 以高概率都存在多项式时间可计算的标准标记。我们的算法将标准的颜色细化程序与基于经典线性时间树标准化算法的简单后处理相结合。值得注意的是，我们对颜色细化在此场景下性能的分析，使我们能够完整描述 $G(n,p)$ 的 2-核的自同构群。